1
Solved example of express in terms of sine and cosine
$\frac{1-\tan\left(x\right)}{1+\tan\left(x\right)}$
2
Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
$\frac{1+\frac{-\sin\left(x\right)}{\cos\left(x\right)}}{1+\tan\left(x\right)}$
3
Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
$\frac{1+\frac{-\sin\left(x\right)}{\cos\left(x\right)}}{1+\frac{\sin\left(x\right)}{\cos\left(x\right)}}$
4
Combine all terms into a single fraction with common denominator
$\frac{\frac{\cos\left(x\right)-\sin\left(x\right)}{\cos\left(x\right)}}{1+\frac{\sin\left(x\right)}{\cos\left(x\right)}}$
5
Combine all terms into a single fraction with common denominator
$\frac{\frac{\cos\left(x\right)-\sin\left(x\right)}{\cos\left(x\right)}}{\frac{\cos\left(x\right)+\sin\left(x\right)}{\cos\left(x\right)}}$
$\frac{\left(\cos\left(x\right)-\sin\left(x\right)\right)\cos\left(x\right)}{\cos\left(x\right)\left(\cos\left(x\right)+\sin\left(x\right)\right)}$
7
Simplify the fraction $\frac{\left(\cos\left(x\right)-\sin\left(x\right)\right)\cos\left(x\right)}{\cos\left(x\right)\left(\cos\left(x\right)+\sin\left(x\right)\right)}$ by $\cos\left(x\right)$
$\frac{\cos\left(x\right)-\sin\left(x\right)}{\cos\left(x\right)+\sin\left(x\right)}$
Final Answer
$\frac{\cos\left(x\right)-\sin\left(x\right)}{\cos\left(x\right)+\sin\left(x\right)}$